[FOM] Does `reduction to set theory' reduce?
John Baldwin
jbaldwin at uic.edu
Thu Dec 27 11:23:11 EST 2007
This a second response to Bill Tait's remarks on Tim Chow's
`formalization thesis' post with a
renamed thread as Bill and I have both veered from Tim original
question.
>On Wed, 26 Dec 2007, William Tait wrote:
> One challenge to the Formalization Thesis, or maybe better, to a
> satisfactory formulation of it, arises from the 'reduction to sets' of
> various types of objects: ordered pairs, functions, ordinal numbers,
> cardinal numbers, real numbers, etc. In each case, the reduction adds
> structure to the type that is not part of its real meaning. In fact,
> this was Dedekind's challenge. (And, as Benacerraf pointed out, there
> is in general some arbitrariness about what structure is added.)
The additional structure imposed by representing a model in set theory can
provide additional information.
1) A fundamental example is Whitehead's
problem.
Whitehead asked whether a group G such that EXT(Z,G) = 0 has to be free.
There is no apparent appeal to cardinality in the statement. But in fact
Shelah proved the result is independent of ZFC for groups of power
aleph_1. A key aspect of his proof is to study a group G whose universe
is aleph_1. Write G as a continuous increasing chain of countable
subgroups G_i and let E be the set of limit ordinals such that G_delta is
not aleph_1 pure in G. Then (ZFC) G is free iff and only if E is not
stationary.
(See Whitehead's Problem is Undecidable
Paul C. Eklof
The American Mathematical Monthly, Vol. 83, No. 10. (Dec., 1976), pp.
775-788.)
E is stationary for every Whitehead group under MA but not under V=L.
Thus the group theoretic property of freeness is dependent on the
underlying set theory.
2) This theme repeats itself in the study of models of first order
theories (without any dependence on axioms beyond ZFC). The proof that (e.g.
unstable theories) have the maximal number of models in every uncountable
power proceeds by representing models as the Ehrenfeucht Mostowski hull of
ordered sets and then analyzing the orders using techniques of stationary
sets.
John T. Baldwin
Director, Office of Mathematics Education
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
312-413-2149
Room 327 Science and Engineering Offices (SEO)
851 S. Morgan
Chicago, IL 60607
Assistant to the director
Jan Nekola: 312-413-3750
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